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The No-Clash Teaching Dimension is Bounded by VC Dimension

Jiahua Liu, Benchong Li

Abstract

In the realm of machine learning theory, to prevent unnatural coding schemes between teacher and learner, No-Clash Teaching Dimension was introduced as provably optimal complexity measure for collusion-free teaching. However, whether No-Clash Teaching Dimension is upper-bounded by Vapnik-Chervonenkis dimension remains unknown. In this paper, for any finite concept class, we construct fragments of size equals to its Vapnik-Chervonenkis dimension which identify concepts through an ordered compression scheme. Naturally, these fragments are used as teaching sets, one can easily see that they satisfy the non-clashing condition, i.e., this open question is resolved for finite concept classes.

The No-Clash Teaching Dimension is Bounded by VC Dimension

Abstract

In the realm of machine learning theory, to prevent unnatural coding schemes between teacher and learner, No-Clash Teaching Dimension was introduced as provably optimal complexity measure for collusion-free teaching. However, whether No-Clash Teaching Dimension is upper-bounded by Vapnik-Chervonenkis dimension remains unknown. In this paper, for any finite concept class, we construct fragments of size equals to its Vapnik-Chervonenkis dimension which identify concepts through an ordered compression scheme. Naturally, these fragments are used as teaching sets, one can easily see that they satisfy the non-clashing condition, i.e., this open question is resolved for finite concept classes.
Paper Structure (7 sections, 4 theorems, 11 equations, 6 figures)

This paper contains 7 sections, 4 theorems, 11 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Concept classes and VC dimension
  4. Machine Teaching and the No-Clash Model
  5. Labeled sample compression scheme
  6. Upper Bounding NCTD with VC Dimension
  7. Conclusion

Key Result

Lemma 1

For each concept class $\mathcal{C}$ of finite functions with $\mathit{VCdim}(\mathcal{C}) = d$ and $|\mathcal{X}| = n$, we have where the number of items in the right side of the inequality is $\binom{n}{d}$.

Figures (6)

  • Figure 1: $\mathcal{C}_1$.
  • Figure 2: Frequencies of fragments in the first round for $\mathcal{C}_1$.
  • Figure 3: Frequencies of fragments in the second round for $\mathcal{C}_1$.
  • Figure 4: Frequencies of fragments in the third round for $\mathcal{C}_1$.
  • Figure 5: Frequencies of fragments in the fourth round for $\mathcal{C}_1$.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Example 1
  • Definition 1: ShinoharaMiyano1991GoldmanKearns1995
  • Definition 2: KuzminWarmuth2007
  • Definition 3: Kirkpatricketal2019
  • Lemma 1
  • proof
  • Example 2
  • Lemma 2
  • proof
  • Theorem 1
  • ...and 5 more